Fundamental Terms and Ideas

Ballot
A ballot is a record of how a voter voted.

 

Majority
A candidate with a majority of the votes has more than half of the votes.
It takes 26 or more votes to make a majority when there are a total of 50 votes.

 

Plurality
A candidate with a plurality of the votes has more votes than any other candidate.
In a 3 candidate election, candidate A gets 14 votes, candidate B gets 20 votes, and candidate C gets 16 votes (total of 50 voters). Candidate B has a plurality of the votes (but, in this case, not a majority of the votes).

Preference Ballot
A preference ballot is an individual voter's ballot for which the voter lists each candidate (or alternative) in order of preference from first to last.
Below is the preference ballot of an individual voter who voted for candidate B for 1st place, candidate C for 2nd place, candidate A for 3rd place, and candidate D for 4th place.
1st : B
2nd: C
3rd: A
4th: D
Notice that a preference ballot is different from a typical U.S. election ballot. On a typical U.S. election ballot, the voter only votes for his/her first choice. However, a preference ballot always gives a complete ranking of the candidates from first to last place.

Suppose a 10 person committee elects a chairman by first casting preference ballots. (Actually only the 6 members who are not running for chairman get to vote.) Following is a listing of the 6 preference ballots (G=George, H=Holly, I=Inez, J=James):
Each of the 6 columns in the body of the table (the unshaded part) represents a preference ballot for a voter. For instance, Carrie's preference ballot is G, H, J, I (meaning that Carrie ranked George first, Holly second, James third, and Inez fourth).

Voters

Place

 Al Betty Carrie Dave Edith Frank
1st G I G G J I
2nd H H H H G H
3rd J J J J H J
4th I G I I I G

THIS IS A SET OF 6 PREFERENCE BALLOTS.

 

Preference Schedule
A preference schedule is a table which summarizes the results of all the individual preference ballots for an election.
Look back at the 6 preference ballots in the example just above. Notice that 3 voters (Al, Carrie, and Dave) cast identical preference ballots (G, H, J, I), 2 other voters (Betty and Frank) cast the same preference ballot (I, H, J, G) and 1 voter (Edith) cast the preference ballot (J, G, H, I).
A preference SCHEDULE summarizes the election results as follows:

# of Voters

  

Place

 3 2 1
1st G I J
2nd H H G
3rd J J H
4th I G I
THIS IS A PREFERENCE SCHEDULE.
Notice that the preference schedule compactly summarizes the votes in the election by grouping identical preference ballots together along with a count of how many voters cast each identical preference ballot.

 

IMPORTANT IDEA
An entire class of voting methods is based on the use of a preference schedule. Not surprisingly, voting methods in this class are called preferential voting methods. In this course, we will explore 4 popular preferential voting methods.

Back to the INTRO TO THE MATHEMATICS OF VOTING